Abstract
A partition ? of the set [n] = {1, 2,...,n} is a collection {B1,...,Bk} of nonempty disjoint subsets of [n] (called blocks) whose union equals [n]. Suppose that the subsets Bi are listed in increasing order of their minimal elements and ? = ?1, ?2...?n denotes the canonical sequential form of a partition of [n] in which iEB?i for each i. In this paper, we study the generating functions corresponding to statistics on the set of partitions of [n] with k blocks which record the total number of positions of ? between adjacent occurrences of a letter. Among our results are explicit formulas for the total value of the statistics over all the partitions in question, for which we provide both algebraic and combinatorial proofs. In addition, we supply asymptotic estimates of these formulas, the proofs of which entail approximating the size of certain sums involving the Stirling numbers. Finally, we obtain comparable results for statistics on partitions which record the total number of positions of ? of the same letter lying between two letters which are strictly larger.
Highlights
A partition Π of the set [n] = {1, 2, . . . , n} is a collection {B1, B2, . . . , Bk} of nonempty disjoint subsets of [n] whose union equals [n]
We assume that B1, B2, . . . , Bk are listed in increasing order of their minimal elements, that is, min B1 < min B2 < · · · < min Bk
Having drawn the graph representation of a partition π of [n], we say that the two points (j, i) and (j, i ) lying on the vertical line x = j have j-distance m if there are m points in the interior of the subset of the first quadrant of Z2 bounded by the line segment between (j, i) and (j, i ) and the horizontal lines emanating in the positive direction from these points
Summary
Enumerating set partitions by the number of positions between. If Π = {1, 4}, {2, 5, 7}, {3}, {6} is a partition of [7], its canonical sequential form is π = 1231242 and in such a case we write Π = π. For other statistics on finite set partitions, see, e.g., [2, 5, 9, 12]. A set A of points in the first quarter of the lattice Z2 is a graph representation for a member of P (n, k) if A contains only points of the form (j, i) such that j ≤ i, j = 1, 2, . We provide asymptotic estimates of these formulas, the proofs of which entail finding the approximate size of certain sums involving the Stirling numbers
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
